| 17 | | [[CollapsibleStart(The Jeans Condition)]] |
| 18 | | Jeans analysis of the linearized 1D gravitohydrodynamic equations of equations an infinite medium |
| | 17 | [[CollapsibleStart(The Jeans Condition/TrueLove Condition)]] |
| | 18 | Jeans' analysis of the linearized 1D gravitohydrodynamic (GHD) equations of a medium of infinite extent lead to the expression for the jeans length: |
| | 19 | |
| | 20 | {{{#!latex |
| | 21 | $ \lambda_j = (\frac{\pi c_s^2}{G \rho})^{1/2}$ |
| | 22 | }}} |
| | 23 | |
| | 24 | which showed that perturbations that are larger than this are gravitationally unstable and collapse. Truelove ('97) showed that these instabilities can be triggered from numerical errors of the GHD solvers. Errors introduced on the cell scale at coarser grids can be transmitted to finer levels and can lead to 'artificial fragmentation'. A way to avoid this is to maintain resolution of the local jeans length. By defining: |
| | 25 | |
| | 26 | {{{#!latex |
| | 27 | $ J = \frac{\triangle x}{\lambda_j} $ |
| | 28 | }}} |
| | 29 | |
| | 30 | Truelove ('97) showed that keeping |
| | 31 | |
| | 32 | {{{#!latex |
| | 33 | $ J \leqq 0.25$ |
| | 34 | }}} |
| | 35 | |
| | 36 | prevented artificial fragmentation of an isothermal cloud spanning 7 decades of density. It is expected to be necessary albeit not necessarily sufficient for isothermal collapse. By not having the resolution of gradients can trigger artificial viscosity (used for numerical stability), which can lead to an incorrect formulation of the problem originally deemed inviscid. |
| | 37 | |
